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real-rootedness)","region_exact":null,"edges":["e-037c64010922a7af"]},{"id":191629,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-037c64010922a7af"]}],"region_exact":{},"incoming_edges":[{"id":"e-037c64010922a7af","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8a263f617dd30ab3","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"f99731eadffe7d08de8da6282bce96ebad8112ff651159bb6c88c7d6cfe68a01","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2b620f031a59b07c","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1,          22,             3;\n   2: 1,         222,            90,             4;\n   3: 1,        2222,          2043,           232,             5;\n   4: 1,       22222,         42150,          9220,           475,            6;\n   5: 1,      222222,        834183,        314360,         29100,          846,           7;\n   6: 1,     2222222,      16182810,       9914324,       1469650,        73836,        1372,          8;\n   7: 1,    22222222,     310806723,     299178152,      66769955,      5160036,      161602,       2080,        9;\n   8: 1,   222222222,    5938661070,    8791397220,    2844461025,    317485602,    14908152,     317808,     2997,     10;\n   9: 1,  2222222222,  113167893663,  254077336920,  116234304450,  18017690922,  1190348229,   37377600,   576315,   4150,   11;\n  10: 1, 22222222222, 2153523312930, 7265055958644, 4618265935800, 968294947752, 86489792004, 3752564376, 84120795, 980650, 5566, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+16*x-x^2)*P(n-2,x) + (72*x-9*x^2)*d/dx P(n-2,x) - (7-2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 22*x + 3*x^2, P(2,x) = 1 + 222*x + 90*x^2 + 4*x^3.","mathematica":"P[1, x_] := 1 + 22*x + 3*x^2;\nP[2, x_] := 1 + 222*x + 90*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[(8 + 16*x - x^2)*P[n - 2, x] + (72*x - 9*x^2)*D[P[n - 2, x], x] - (7 - 2*x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 22*x + 3*x^2; v[2] = 1 + 222*x + 90*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = (8 + 16*x - x^2)*v[i - 2] + (72*x - 9*x^2)*deriv(v[i - 2], x) - (7 - 2*x)*v[i - 1] + 9*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2b620f031a59b07c","steps":["\u03b8+1"],"edges":["e-037c64010922a7af"],"text":"A166979 \u2192 \u03b8+1"},"headline":"A166979 \u2192 \u03b8+1","chain":null}